A generalized Anderson single-impurity model with off-site Coulomb interactions is derived from the extended three-band Hubbard model, originally proposed to describe the physics of the copper oxides. Using the Abelian bosonization technique and canonical transformations, an effective Hamiltonian is derived in the strong-coupling limit, which is essentially analogous to the Toulouse limit of the ordinary Kondo problem. In this limit, the effective Hamiltonian can be exactly solved, with a mixed-valence quantum critical point separating two different Fermi-liquid phases, i.e., the Kondo phase and the empty orbital phase. In the mixed-valence quantum critical regime, the local moment is only partially quenched and x-ray edge singularities are generated. Around the quantum critical point, a type of non-Fermi-liquid behavior is predicted with an extra specific heat C-imp similar to T-1/4 and a singular spin susceptibility chi(imp)similar to T-3/4. At the same time, the effective Hamiltonian under single occupancy is transformed into a resonant-level model, from which the correct Kondo physical properties (specific heat, spin susceptibility, and an enhanced Wilson ratio) are easily rederived. Finally, a brief discussion is given to relate these theoretical results to observations in UPdxCu5-x (x=1,1.5) alloys, which show single-impurity critical behavior consistent with our predictions.